Question

Use a graphing calculator to graph the function and its parent function. Then describe the transformations. \(f(x)=\frac{3}{4}|x|+1\)

Short answer

The parent function \(|x|\) was transformed by a vertical stretch of factor \(\frac{3}{4}\) and then an upward shift of 1 unit to obtain the function \(\frac{3}{4}|x|+1\).

Step-by-step solution

Graph parent function \(|x|\)

Begin by graphing the parent function \(|x|\). In a graphing calculator, this would involve simply entering the function and observing the graph. The graph of \(|x|\) is a straight line that passes through the origin (0,0) and opens upwards in a V shape.

Apply vertical stretch to parent function

Next, apply a vertical stretch of \(\frac{3}{4}\) to the parent function to get \(\frac{3}{4}|x|\). To graph this, multiply each y-coordinate in the parent function by \(\frac{3}{4}\). This transformation results in a vertical stretch that makes the graph thinner by a factor of \(\frac{3}{4}\).

Apply vertical shift to stretched parent function

After doing the vertical stretch, apply a vertical shift of 1 unit by adding 1 to every y-value in the stretched function. This means that every point on the graph of \(\frac{3}{4}|x|\) will be moved one unit upwards. So the transformed function becomes \(\frac{3}{4}|x|+1\).

Final graph and description of transformations

Finally, the graph of \(f(x)=\frac{3}{4}|x|+1\) is obtained. The transformations that occured are: a vertical stretch by a factor of \(\frac{3}{4}\) and an upward translation or shift by 1 unit.

Key concepts

Parent Function

Understanding the concept of parent functions is critical to grasp transformations in algebra. A parent function is the simplest form of a function family and sets the stage for more complex variations through transformations. For instance, the absolute value function, denoted as |x|, is a common parent function. It is graphically represented by a V-shaped graph that passes through the origin \(0,0\) and extends upwards and outwards.

The knowledge of a parent function's shape and equation is essential, as all transformations can be visualized in terms of changes to this base function.

Vertical Stretch

When we talk about a vertical stretch, imagine pulling a graph away from the x-axis.

What is a Vertical Stretch?

A vertical stretch is a transformation that either elongates or compresses the graph of a function vertically. Mathematically, it is applied by multiplying the function's output by a factor more than one to stretch, or between zero and one to compress. In the transformation \( \frac{3}{4}|x| \), we see this factor is \( \frac{3}{4} \), meaning each point of the parent function is reduced to \( \frac{3}{4} \) of its original distance from the x-axis. This results in a graph that's compressed rather than stretched.

Vertical Shift

Moving on to vertical shifts, they slide the graph up or down along the y-axis.

Understanding Vertical Shifts

A vertical shift repositions a function's graph along the y-axis without altering its shape or symmetry. This transformation is performed by adding (to shift upwards) or subtracting (to shift downwards) a constant value to every point's y-coordinate. Looking at the function \( \frac{3}{4}|x|+1 \), the '+1' indicates an upward shift by 1 unit. Consequently, every point of the already vertically compressed graph moves one unit higher.

Absolute Value Function

Diving deeper into the absolute value function, we see its unique characteristics.

Properties of the Absolute Value Function

An absolute value function, symbolized as \( |x| \), reflects the distance of a number on the number line from zero without considering direction, which is why its value is always non-negative. This function is notable for its critical point at the origin, where the graph changes direction, creating the distinctive V shape. By understanding this core shape, it becomes easier to predict the effects of transformations like stretches and shifts.

Graphing Calculator Usage

A graphing calculator is an essential tool for visualizing function transformations.

How Does a Graphing Calculator Help?

By inputting the equation of a function into a graphing calculator, students can immediately see the graph and analyze changes as transformations are applied. For example, it can quickly display the parent function \( |x| \), then show the effect of multiplying it by \( \frac{3}{4} \), and finally, exhibit the graph after shifting it up by 1 unit. Mastering the use of a graphing calculator empowers students to explore and verify their understanding of transformations without manual plotting.